1. Field of the Invention
This invention relates to a methodology for preventing dip-induced spatial aliasing of the 3-D Kirchhoff DMO operator applied to data acquired along random trajectory azimuths.
2. Discussion of Related Art
As is well known to geophysicists, a sound source, at or near the surface of the earth, is caused periodically to inject an acoustic wavefield into the earth at each of a plurality of regularly-spaced survey stations. The wavefield radiates in all directions to insonify the subsurface earth formations whence it is reflected back to be received by seismic sensors (receivers) located at designated stations at or near the surface of the earth. The seismic sensors convert the mechanical earth motions, due to the reflected wavefield, to electrical signals. The resulting electrical signals are transmitted over a signal-transmission link of any desired type, to instrumentation, usually digital, where the seismic data signals are archivally stored for later processing. The travel-time lapse between the emission of a wavefield by a source and the reception of the resulting sequence of reflected wavefields by a receiver, is a measure of the depths of the respective earth formations from which the wavefield was reflected.
The seismic survey stations of a 3-D survey are preferably distributed in a regular grid as in FIG. 1 over an area of interest with inter-station grid spacings, dx and dy on the order of 25 meters. The processed seismic data associated with a single receiver are customarily presented as a one-dimensional time scale recording displaying rock layer reflection amplitudes as a function of two-way wavefield travel time. A plurality of seismic traces from a plurality of receivers sequentially distributed along a line of survey such as 10 or 12 may be formatted side-by-side to form an analog model of a cross section of the earth (two-dimensional tomography). Seismic sections from a plurality of intersecting lines such as 10 and 12 of survey distributed over an area of interest, provide three-dimensional tomography.
To provide some definitions, the term "signature" as used herein means the variations in amplitude and phase of an acoustic wavelet (for example, a Ricker wavelet) expressed in the time domain as displayed on a time scale recording. The impulse response means the response of the instrumentation (source, receivers, data-processor, transmission link, earth filter, etc.) to a spike-like Dirac function. To prevent aliasing, the sample interval (spatial or temporal) must be less than half the signal period. DMO (dip moveout) is an imaging operator that offers a stratagem for correcting the reflection-point smear that results from stacking of dipping reflections.
If we can assume constant velocity, the 3-D Kirchhoff DMO operator is simply the 2-D DMO operator constructed along an arbitrary source-receiver trajectory. In FIG. 1. the line segment 10 containing the data, lying between the source, S.sub.1, and the receiver (sensor), R.sub.1, is the DMO aperture along that azimuth. Line 12 represents a DMO aperture lying along some other randomly-chosen azimuth. The grid-line intersections represent the stack output points or bins. The output grid points are characterized by biaxial, preferably but not limited to, orthogonal boundaries of preselected dimensions.
Theoretically, the aperture is a continuum. But in order to stack the impulse responses, the continuous aperture must be discretized in surface coordinates x and y as shown in FIG. 1. Conventionally that is done by stacking discretely sampled points along the aperture at the nearest output grid point. If the source-receiver azimuth is not favorably oriented in relation to the 3-D grid, spatial aliasing of the DMO operator results. In FIG. 2, 14 is an example of a conventionally discretized DMO aperture. Observe the stair-stepping distortion. FIG. 3 is the 2-D spectrum of the DMO aperture shown in FIG. 2 for -.pi.&lt;kx,ky&lt;+.pi., demonstrating the wrap-around aliasing thereof,where kx and ky are the wavenumbers in the x and y coordinates.
Early students of DMO application to 3-D geometries along arbitrary azimuths include Vermeer et al. and Cooper et al. Vermeer, in a paper entitled DMO in arbitrary 3-D acquisition geometries, presented at the 66th meeting of the Society of Exploration Geophysicists (1995) and published as Expanded Abstract paper, pp 1445-1448 showed that for cross spread geometry, the locus of contributing midpoints for a given output point is a hyperbola in the (x,y) plane passing through the output point, provided the data are sampled alias-free. However in actual practice, according to Vermeer, correct, alias-free sampling of the hyperbolas is difficult to achieve. Even in regularly sampled data, the result of 3-D DMO is suboptimal.
The following year, Cooper et al, at the 1996 meeting of the Society of Exploration Geophysicists delivered a paper entitled 3-D DMO for cross spread geometry: a practical approach in multifold-field-data, which is published in the Expanded Abstracts, pp 1483-1486. Cooper follows the pathway marked by Vermeer except that he spatially over-samples the input data prior to "smiling" the data along the source-receiver azimuth. This technique results in improved signal-to-noise ratio but at a very considerable increase in computational costs due to the over sampling and the finer resolution of the "smiles".
There is a need for a practical, computationally efficient method for spatial de-aliasing of the 3-D Kichhoff DMO operator.